Interpolation sequences for the Bergman space
The main results of the paper. Theorem A. Suppose \(A = \{a_ n\}\) is a sequence of interpolation for \(L^ 2_ a (\mathbb{D})\) -- the Bergman Space. Then there exists a unique sequence \(\{t_ n\}\) in \(L^ 2_ a (\mathbb{D})\) such that the kernel function \(K_ A (z,w)\) admits the following partial fraction expansion \[ K_ A (z,w) = (1-z \overline w)^{-2} - \sum^ \infty_{n=1} (1-\overline a_ nz)^{-2} \overline {\psi_ n (w)}, \] \(z,w \in \mathbb{D}\). Moreover (1) \(\psi_ n (a_ n) = 1\) and \(\psi_ n (a_ m) = 0\) for all \(n,m \geq 1\) and \(n \neq m\). (2) For each compact set \(K\) in \(\mathbb{D}\) there exists a positive constant \(C_ k\) such that \[ \bigl | \psi_ n (z) \bigr | \leq C_ k \bigl( 1-| a_ n |^ 2 \bigr)^{3/2},\;z \in K,\;n \geq 1. \] (3) There is a constant \(C>0\) such that \(1-| a_ n |^ 2 \leq \| \psi_ n \| \leq C (1-| a_ n |^ 2)\) for all \(n \geq 1\). Theorem B. Suppose that \(A = \{a_ n\}\) is a sequence of interpolation for \(L^ 2_ a (\mathbb{D})\) and that \(\{\psi_ n \}\) is the sequence from Theorem A. If \(\{w_ n\}\) is a sequence of complex numbers satisfying \(\sum^ \infty_{n=1} (1 - | a_ n |^ 2)^ 2 | w_ n |^ 2 < + \infty\), then the series \(\sum^ \infty_{n=1} w_ n \psi_ n (z)\) converges to a function in \(L^ 2_ a (\mathbb{D})\) which uniquely solves the minimal interpolation problem \(\inf \{\| f \|:f(a_ n) = w_ n,\;n \geq 1\}\).
scientific article; zbMATH DE number 575621